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Unformatted text preview: STA 302 / 1001 F  Fall 2005 Test 2 November 16, 2005 LAST NAME: SOLUTIONS FIRST NAME: STUDENT NUMBER: ENROLLED IN: (circle one) STA 302 STA 1001 INSTRUCTIONS: • Time: 90 minutes • Aids allowed: calculator. • A table of values from the t distribution is on the second to last page (page 9). • A table of formulae is on the last page (page 10). • For all questions you can assume that the formulae on page 10 are known. • Total points: 44 1 2ab 2cd 3abc 3d 3efg 4 7 7 10 3 9 1 1. (4 points) Suppose that X is a 2 × 1 random vector with E( X ) = 1 2 ! and Cov( X ) = 4 1 1 9 ! . Y is another random vector with Y = AX where A is the constant matrix A = 12 3 ! . Find the expectation of Y and the variancecovariance matrix for Y . E( Y ) = A E( X ) = 12 3 ! 1 2 ! =3 6 ! Cov( Y ) = A Cov( X ) A = 12 3 ! 411 9 ! 12 3 ! = 6193 27 ! 12 3 ! = 445757 81 ! 2 2. (14 points) (a) Write the simple linear regression model in matrix terms, defining all terms. (3 marks) Y = X β + ² where Y = Y 1 Y 2 . . . Y n β = β β 1 ! X = 1 X 1 1 X 2 . . . . . . 1 X n ² = ² 1 ² 2 . . . ² n (b) Explain why Cov( ² ) = σ 2 I follows from the assumptions of simple linear regression. (4 marks) Cov ( ² ) is the n × n matrix with i th diagonal entry equal to the variance of ² i and ij th off diagonal entry equal to Cov ( ² i , ² j ) . Since two regression assumptions are that Var ( ² i ) = σ 2 for all i and Cov ( ² i , ² j ) = 0 , it follows that Cov ( ² ) = σ 2 I ....
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This note was uploaded on 09/27/2009 for the course STA STA302 taught by Professor Gibbs during the Fall '04 term at University of Toronto.
 Fall '04
 Gibbs

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